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Quantum toroidal algebras and quantum integrable systems

Research Project

Project/Area Number 19K03549
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12010:Basic analysis-related
Research InstitutionRikkyo University

Principal Investigator

JIMBO Michio  立教大学, 名誉教授, 名誉教授 (80109082)

Project Period (FY) 2019-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywordsquantum toroidal algebra / deformed W algebra / integrals of motion / 量子トロイダル代数 / 運動の保存量 / リー超代数 / シャッフル代数 / トロイダル量子群 / W代数 / qq指標 / スクリーニング作用素 / qq 指標 / qq character / 共形場理論
Outline of Research at the Start

従来の量子可積分系研究はほとんどがアフィン量子群の対称性を基盤とするものであった。本研究はその手法をトロイダル量子群対称性をもった量子可積分系に拡張することを目的とする。具体的には共形場理論における運動の保存量のq変形を対象とし、表現論的な方法に基づいて可換作用素族の構成とそのスペクトルの記述を行う。特に懸案であるODE/IM対応(スペクトルとある種の微分作用素族との間の1対1対応)の解明を目標とする。

Outline of Final Research Achievements

We studied quantum toroidal algebras in view of its applications to integrable systems. We obtained the following results: 1)We determined the branching rule of Wakimoto representations of quantum toroidal gl_n to its subalgebra which is a product of gl_1 quantum toroidal algebras. This shows explicitly that the deformation of the coset W algebra of type gl_n/gl_{n-1} is given by deformed W superalgebra W(gl_{n|n-1}). 2) We introduced an algebra K_1, which is a comodule over gl_1 quantum toroidal algebra, gave a uniform description of deformed W algebras of classical types, and constructed a commutative subalgebra (integrals of motion) thereof. 3) We generalized the algebra K_1 to a gl_n analog K_n, and constructed its commutative subalgebra.

Academic Significance and Societal Importance of the Research Achievements

W代数は共形場理論の数学的定式化である。量子トロイダル代数はW代数のq変形の研究に有力な方法を与えている。これまでの研究は概ねA型の場合に限られていたが、本研究では量子トロイダル代数を少し拡張することによって、一般の場合の統一的扱いに一歩を踏み出した。特に「localな運動の保存量」と呼ばれる可換な部分代数の構成がA型の場合とほぼ同じ方法でできることがわかり、今後の研究への素材を提供できたと考えている。

Report

(5 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (10 results)

All 2022 2021 2019

All Journal Article (7 results) (of which Int'l Joint Research: 7 results,  Peer Reviewed: 7 results) Presentation (2 results) (of which Int'l Joint Research: 2 results,  Invited: 2 results) Book (1 results)

  • [Journal Article] Combinatorics of vertex operators and deformed W-algebra of type D(2,1;alpha)2022

    • Author(s)
      B.Feigin, M.Jimbo, and E.Mukhin
    • Journal Title

      Adv. Math.

      Volume: 403 Pages: 108331-108331

    • DOI

      10.1016/j.aim.2022.108331

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Quantum Toroidal Comodule Algebra of Type A_{n-1} and Integrals of Motion2022

    • Author(s)
      Feigin Boris、National Research University Higher School of Economics, Russia、Jimbo Michio、Mukhin Evgeny、Rikkyo University, Japan、Indiana University Purdue University Indianapolis, USA
    • Journal Title

      Symmetry, Integrability and Geometry: Methods and Applications

      Volume: 18 Pages: 051-051

    • DOI

      10.3842/sigma.2022.051

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Combinatorics of vertex operators and deformed W-algebra of type D(2,1;\alpha)2022

    • Author(s)
      B.Feigin, M.Jimbo and E.Mukhin
    • Journal Title

      Advances in Mathematics

      Volume: 403

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Deformations of W algebras via quantum toroidal algebras2021

    • Author(s)
      Feigin B., Jimbo M.,Mukhin E., Vilkoviskiy I.
    • Journal Title

      Selecta Mathematica

      Volume: 27 Issue: 4

    • DOI

      10.1007/s00029-021-00663-0

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Evaluation modules for quantum toroidal gl_n algebras2021

    • Author(s)
      B.Feigin, M.Jimbo and E.Mukhin
    • Journal Title

      Progress in Mathematics

      Volume: 337

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The (gl_m,gl_n) duality in the quantum toroidal setting2019

    • Author(s)
      B.Feigin, M.Jimbo and E.Mukhin
    • Journal Title

      Commun. Math. Phys.

      Volume: 367 Issue: 2 Pages: 455-481

    • DOI

      10.1007/s00220-019-03405-8

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Towards trigonometric deformation of sl_2 coset VOA2019

    • Author(s)
      B.Feigin, M.Jimbo, and E.Mukhin
    • Journal Title

      J. Math. Phys.

      Volume: 60 Issue: 7 Pages: 073507-073507

    • DOI

      10.1063/1.5081799

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Deforming integrals of motion via quantum toroidal algebras2019

    • Author(s)
      Michio Jimbo
    • Organizer
      Geometry and Integrable Systems
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Remarks on deformed W algebras and integrals of motion2019

    • Author(s)
      Michio Jimbo
    • Organizer
      New Trends in Integrable Systems
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Book] Local Operators in Quantum Integrable Models vol.I2021

    • Author(s)
      M.Jimbo, T.Miwa, and F.Smirnov
    • Total Pages
      192
    • Publisher
      AMS
    • ISBN
      9781470465520
    • Related Report
      2021 Research-status Report

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Published: 2019-04-18   Modified: 2024-01-30  

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